HOMOTOPY GROUPS OF JOINS AND UNIONS BY GEORGE W. WHITEHEAD 1. Introduction. The homology groups of the join X * Y of two spaces have long been known to satisfy a "Kiinneth formula." On the other hand, little is known about the homotopy groups of X * Y. There is a bilinear pair- ing (a, B)—>a* B of iTp(X) with irq( Y) to irp+q+i(X * Y); a * 8 is called the join of a and B. If Y=Sn is an w-sphere and i generates ir„( Y), then it follows from the Freudenthal suspension theorem that the map ct—>a* i is, for small p, an isomorphism of tp(X) with ttp+„+i(X * Y). Thus, in low dimensions, the homo- topy groups of X * Sn are generated by the joins of the homotopy groups of the factors. In this paper we consider the homotopy groups of the join of X with an arbitrary 1-connected CW-complex Y. There is a spectral sequence whose initial term is, in low dimensions, the homology group(l) of F with coefficients in the homotopy group(1) of X and whose final term is the graded group of the homotopy group of X * Y with respect to a suitable filtration. As a con- sequence, the homotopy groups of X * Y, even in low dimensions, are not, in general, generated by the joins of the homotopy groups of the factors. A further consequence is a new proof of the symmetry of the stable Eilenberg- MacLane groups, which was first proved by H. Cartan [4]. These results can be used to study the homotopy groups of the union X\/ Y of two spaces X and Y with a single point in common. The space X\/Y can be naturally imbedded in XX Y and we have a natural iso- morphism TT„(XV F) « lCn(X) © Tn(Y) © X^ifX X F, X V F). Now there is a homomorphism of 7rB+i(XX Y, XV F) into irn+2(X * Y), which is an isomorphism in low dimensions. Thus we can apply the above results on the homotopy groups of X * Y. In particular, the group 7rn+i(XX F, XV F) is not in general generated by "generalized Whitehead products" of elements of the homotopy groups of X and F, even in low dimensions. In a recent paper [0] M. G. Barratt and J. H. C. Whitehead have intro- duced an exact couple for an arbitrary CW-triad. This exact couple, in the special case of the triad (XV F; X, F), seems to be closely related to the one we have introduced here. However, our results do not seem to follow Presented to the Society, June 19, 1954; received by the editors August 4, 1955. (') By the homotopy {homology) group of a space X, we mean the direct sum of the homotopy (homology) groups in all dimensions. 55 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 56 G. W. WHITEHEAD [September immediately from their work; furthermore we do not assume that X is a CW-complex. 2. Joins. Let X and Y be spaces, which we assume to be disjoint from each other and from XXYXI, where / is the unit interval {<|0^gl}. Let W = X\J(XX YXI)^JY; we topologize Why defining a subset to be open if and only if its intersection with each of the spaces X, XX YXI, and Y is open. The join of X and Y is the identification space X * Y obtained from W by identifying each xEX with all of the points (x, y, 0) and each yE Y with all of the points (x, y, 1). The identification map sends X and Y homeo- morphically into X * Y; hence we may consider X and Y as subspaces of X * Y. Let (1 —i)x®ty be the image of (x, y, t) in X * Y. The join operation is easily seen to be commutative (up to a natural homeomorphism). The join of X with the empty set 0 is X. Let f:X—*X', g: Y—rY' be maps. The join of f and g is the map f*g:X* Y->X'* Y' defined by (f*g)((l ~ Dx © ty) = (1 - t)f(x) © tg(y), (f*g)(x)=f(x), (f*g)(y) = g(y); continuity of f * g follows from the fact that the join has the identification topology. Since X * Y is an identification space of W and since I is compact, it follows [18, Lemma 4] that (X * Y) XI is an identification space of WXI. Hence, if /, f':X—*X' are homotopic and if g, g': Y—+Y' are homotopic, then f*g and /' *g' are homotopic maps of X* Y into X' * Y'. It follows that, if X and X' have the same homotopy type, and if Y and Y' have the same homotopy type, then X * Y and X' * Y' have the same homotopy type. lff-.XCX' and g:YCY', then f*g is a 1:1 map of X* Y into X' * Y'; if furthermore X is closed in X' and Y is closed in Y', then/*g is a homeo- morphism and its image is closed in X' * Y'; in that case we may consider X * Y as a closed subset of X' * Y'. Note that the join of the inclusion map of 0 into X' with any map g\ Y—*Y' is the composition of g with the inclusion map of Y' into X' * Y'. Let P be a fixed space consisting of a single point p. The cone over X is the join X = X *P. Let A be a closed subspace of X; then the quotient space Of X by A is the subspace X-i-A =XVJA of X. The following properties are easily verified: (2.1) X is contractible; (2.2) The triad (X+A ; X, A) is proper. Let P' be a fixed space consisting of two points p+, p— The suspension S(X) of X is the join X * P'. Note that S(X) =X+^JX-, where X± = X* p+, while X+C\X- = X. Thus S(X) is the union of two contractible spaces whose intersection is X. It is easy to see that the triad (S(X); X+, X-) is proper. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1956] HOMOTOPY GROUPS OF JOINS AND UNIONS 57 Let <r, r he ordered Euclidean simplexes with vertices ao< • ■ -<ap, bo< • ■ ■ <bq respectively. Then o*t has>a natural affine structure, and we may regard a * r as an ordered Euclidean simplex with vertices a0< • • . <ap <bo< • • ■ <bq. We consider the empty set as an ordered Euclidean ( — 1)- simplex. With the usual definition of equivalence of singular simplexes, we define SP(X) to be the free abelian group generated by the singular (p — 1)- simplexes in X (p = 0, 1, 2, • • • ) and define S(X) = 23p_0 SP(X), with the usual boundary operator (the boundary of each singular 0-simplex is the unique singular (—"1)-simplex), to be the augmental total singular complex of X. If A EX, then S(A) is a subcomplex of S(X) and we define BP(X, A) to be the (p + l)st homology group of the complex S(X)/S(A). Of course, 8P(X, A) = HP(X, A) (p^O), except that Bo(X, 0) is the reduced O-dimensional homology group of X. Let <f>:a—rX, \(/:t—>Y be singular simplexes; then 0»^:<r*r—>X* Y is a singular simplex. The map d>®\p—xp*^ induces a homomorphism a:S(X) ®S(Y) —rS(X * Y), which is easily seen to be a chain mapping. It is known that a is, in fact, a chain-equivalence(2). Furthermore, a is natural in the sense that, if f'.X—>X' and g: Y—*Y'are maps, and if/', g', (f*g)' are the chain maps induced by/, g, and f*g, then the diagram S(X) ® S(Y)L¥-L> S(X') <2>S(Y') [a [a S(X* Y) f*S >S(X'* Y') is commutative. Hence, if A is a closed subset of Y, a induces a chain-equiva- lence between the complexes S(X)® [S(Y)/S(A)] and S(X* Y)/S(X*A), Hence (2.3) The singular homology groups of (X * Y, X * A) are given by Hg+i(X*Y,X*A) « Y Hi(X) ® Hj(Y, A) © Y Tor {8 t(X), Hj(Y, A)}. i+i=q i+i—q—1 It is also known [13, Lemma 2.2] that (2.4) If X is O-connected and Yy^0, then X * Y is l-connected. Consequently we can conclude from the relative Hurewicz theorem [1,(23.3)]: (2.5) If X is (m — l)-connected (m^l), (Y, A) is (n —1)-connected (n^2), and A is l-connected, then (X * Y, X*A) is (m+n)-connected, and 7rm+B+1(X*F, X*A) « irm(X) ® Tn(Y, A). The group 8p(X)®Hg(Y, A) has a natural imbedding }n Hp+q+2(S(X) (2) This fact does not seem to be stated explicitly in the literature, but is not difficult to deduce from Milnor's proof [13] of the "Runneth theorem" for the homology groups of the join. The author is indebted to Milnor for the opportunity of reading his manuscript.